 This video is going to talk about using substitution to solve systems. In substitution, we want to have one equation that is either y equals something or x equals something. So on one side of the equation, you only have one variable with the coefficient of one. We can then substitute that variable into the other equation. In this first example, we see that x is equal to 2. And remember that a solution to a system, the x has to be the same for both equation as well as the y. It has to satisfy both equations. So if x equals 2 satisfies this equation, that means in the bottom, x equals 2 also satisfies. So I can replace the x with my 2 and then finish out the problem and solve for y. When I use substitution, now I get a one variable equation that I know how to solve. And in fact, 2 plus 5 is going to be 7. And now I know that I have the ordered pair 2 for x and 7 for y. Sometimes you have a y equation like the second example. So when I look at the system and I say that again y is negative 3 for both of them, then I can replace the y in the top equation with the negative 3. So I rewrite my problem x minus and then I have my negative 3. And then that's going to be equal to 10. While solving the equation, x plus 3 will equal 10. And if I subtract 3 from both sides, then I find out that x is equal to 7. And I know the ordered pair x equals 7 and y is negative 3 would be the ordered pair that satisfies the system. In this equation, we have a y equal equation, but it's not equal to a number. It's equal to an expression. But just like we did when we had a number in there, we're going to take our expression and replace it with y. So I have my 4x minus 5, but then I have to replace y with 1 minus 2x. And that's going to be equal to 23. So I have 4x, but remember now here we have to distribute negative 5 into both of these things. So I have negative 5 and negative 5 times negative 2x will be plus 10x and that'll be equal to my 23. So if I combine like terms, I have my 4x and my 10x will give me 14x minus 5 on the left-hand side equal to 23. And I can add my 5 to both sides. So now I have 14x equal to 28. And if I divide by 14, I find out that x is equal to 2. Now I'm not done because remember it's an ordered pair and this wasn't a y equal sum number. It was a y equal expression. But I know what x is, so I'm going to substitute my x back into my y equal 1 minus 2x equation. So y, that's what I'm solving for, minus 1 times equal 1 minus 2 times my x, which is 2. And that says then that y is equal to 1 minus 4 or negative 3. And so the ordered pair is x is 2, y is negative 3. Alright, now what do we do? Now we don't have a y equal or an x equal equation, but I have a y that has a coefficient of 1 on it. So we're going to have to solve for that y before I can even start substituting. So we take and subtract the 5x from both sides. So on the left-hand side we're left with negative y and then we have our 10 minus 5x. And now we have to divide everything by negative 1 so that we have a positive y. So y is going to be equal to negative 10 and a negative times a negative is a plus 5x. I'm going to substitute negative 10 plus 5x into this y right here. Rewriting that equation then, 3x plus 4. And then I have my y that I just found to be negative 10 plus 5x. And that's going to be equal to 6, finishing out the equation. So 3x, but remember now I have to distribute 4 into both of these things. So minus 40 and plus 20x and that'll be equal to my 6. Combining like terms I have 23x minus 40. There's my 3 and my 20. Equal to 6, add 40 to both sides and 23x is going to be equal to 46. And if we divide by 23, x will be, that's a 3, x will be 2. So I have x, but I don't have y. I need to plug that back into this x right here. So y is equal to negative 10 plus 5 times my x that I just found. And that gives me y equal negative 10 plus 10, so y equal 0. And the ordered pair is 2 for the x and 0 for the y. There's a couple of special cases. What happens when we have those either parallel lines which have no solution or one line which has infinite solutions? What does that look like when we do substitution? Well, we need to solve one of these equations for x or for y and it doesn't look like any of them are going to be really, really nice. But let's solve for x. And I want to solve for x in this bottom equation only because it's smaller numbers. And part of the reason I'm solving for x is because it's a positive number. So if I solve for x in this equation, then I'm going to add the 3y to both sides. So 2x is going to be equal to negative 2 plus 3y. And then I have to divide by 2 and we couldn't avoid fractions, could we? x is equal to negative 2 over 2 is negative 1 plus, and go ahead, you might prefer to put 1.5y in here since that's a little bit easier to substitute. Here's what I'm going to substitute. And I'm not going to have, I have to put it into the other equation so I'm going to plug it in for this x. So let's rewrite our equation. 8, but now I'm going to replace my x and then minus 12y is equal to negative 8. And if I put my x in here now, it's negative 1 plus 1.5y. Distributing, I'm going to have negative 8 plus 12y minus 12y equal to negative 8. Well, these are opposites of each other and my like terms are opposites of each other so they cancel each other out and I'm just left with negative 8 equal negative 8. Well, I don't have any variables, but we've done this, we've looked at these kinds of things before. If I have a negative 8 equal to a negative 8, that's a true statement. And if it's a true statement, it means it's always true. And if it's always true, it must mean that I have the same line or infinite solutions if you would rather write it that way. That's the case number two. This one will be a little bit easier because we've got a y here that we can solve for very easily. I'm going to subtract 4x from both sides so that top equation becomes y equal 3 minus 4x. So I've got my 3 minus 4x and I'm going to plug it in for y and the y of the other equation. So when I rewrite that problem, I have 8x is equal to 12 minus 2 and then I'm not going to write y, I'm going to write 3 minus 4x. So working through the problem, 8x is equal to 12 and then I'm going to distribute my negative 2 inside everything here. So negative 6 plus 8x. And if I combine my light terms, I have 8x on the left-hand side equal to 6 plus 8x. And if I subtract my 8x from both sides, I get 0 on this side because they cancels each other out which is false. And a false statement means that we can have no solution. If we were to graph those, they would be a set of parallel lines.